Simplex Method Calculator with Slack Variables
Simplex method calculator with slack variables: converts <= inequalities into equations with slack variables and solves the LP step by step.
Simplex Calculator
How Simplex Method Calculator Works
Enter the LP Problem
Type the objective function coefficients and every constraint row with its right-hand-side value.
Choose Maximize or Minimize
Pick your optimization goal. The tool builds the initial tableau with slack variables automatically.
Run the Pivot Iterations
The calculator identifies pivot column by Cj-Zj, computes ratios, performs elementary row operations until optimal.
Read the Optimal Solution
Final tableau displays optimal variable values, Zj row, and the maximum/minimum objective value.
Sample Simplex Tableau Output
Example tableau iteration for a 2-variable maximization problem
| Basis | x1 | x2 | s1 | s2 | RHS | Cj-Zj |
|---|---|---|---|---|---|---|
| x1 | 14 | 0 | 0 | 1 | 14 | 0 |
| x2 | 7 | 1 | 0 | 0 | 7 | 5 |
| Zj | 35 | 5 | 0 | 0 | 35 |
See the Standard Form Before the Tableaus
Before the first tableau, the calculator prints the standard form of your problem: every constraint rewritten as an equation with its slack variable (for <= constraints), surplus and artificial variables (for >= constraints) or artificial variable (for = constraints), and the objective with these variables added. Each helper variable is labelled, for example s1 = slack variable, so you can check your own conversion line by line. For every pivot it also lists the row operations, such as R2 -> R2 - 8 x R1, used to build the next tableau.
What Slack Variables Do
A slack variable is added to a less-than-or-equal constraint to turn the inequality into an equation. For example, x1 + x2 <= 5 becomes x1 + x2 + s1 = 5, where s1 is the unused capacity. This simplex method calculator with slack variables adds one slack per constraint, uses them as the starting basic feasible solution, and displays their columns in every tableau.
Slack, Surplus and Artificial Variables
Less-than constraints add slack variables, greater-than constraints subtract surplus variables, and equality or greater-than constraints add artificial variables. The calculator handles all three automatically so any linear program can be put into standard form and solved.
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Frequently Asked Questions
What is a slack variable?
A slack variable is a non-negative variable added to a less-than-or-equal constraint to convert it into an equation, representing the unused amount of that resource.
Why are slack variables needed in the simplex method?
They convert inequality constraints into equations and provide an initial basic feasible solution (the identity columns) to start the algorithm.
What is the difference between slack and surplus variables?
A slack variable is added to a less-than constraint; a surplus variable is subtracted from a greater-than constraint to make it an equation.
Do slack variables appear in the final solution?
Yes. A basic slack variable that is positive shows leftover capacity in that constraint; a slack of zero means the constraint is binding.
How many slack variables are added?
One slack variable is added for each less-than-or-equal constraint in the problem.